Shape Dynamics is a fairly recent approach which might someday lead to a quantum theory of gravity. The shape dynamics formulation of gravity possesses a physical Hamiltonian that generates evolution of spatial conformal geometry. The theory is constructed to give the same predictions as Einstein's theory of gravity, on a local scale.  The basic idea is that this more general formulation might be easier to quantize than Einstein's inherently classical field theory.  This approach is only a decade old, and it is difficult to estimate how likely it is to progress further.

• Quantum Gravity in String Theory--- the description is perturbative and background dependent. There is no known way to get from the string theory landscape to our actual universe. The duality between various string theory descriptions of types of universes (and black holes) with various conformal field theories could someday give a hint as to what a usable theory of quantum gravity might be like.

• Loop Quantum Gravity--- this approach is not even consistent with special relativity since it treats time differently than space, quantizing only space. The theory makes no testable predictions that could lead to modifications bringing it into agreement in the classical limit with Einstein's Theory of Gravity.

• Causal Dynamic Triangulation and Causal Sets--- Both of these approaches quantize space-time directly. The more general of the two is Causal Set theory. Both approaches abandon the quest for an analytic mathematical description, and rely on purely numerical simulations. Both approaches have difficulty in making a connection to the phenomena of the actual universe.

There is no shortage of new ideas for tackling the quantization of gravity, but none of the approaches that I am aware of are free of possible crippling limitations or obstacles.